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How many solutions does the system of equations below have? {:[-10 x+3y=-8],[20 x-6y-16=0]:} (A) no solution (B) one solution (C) infinitely many solutions

How many solutions does the system of equations below have?\newline−10x+3y=−820x−6y−16=0 \begin{array}{l} -10 x+3 y=-8 \\ 20 x-6 y-16=0 \end{array} \newline(A) no solution\newline(B) one solution\newline(C) infinitely many solutions

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Determine the number of solutions to a system of equations in three variablesright chevron icon

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Q. How many solutions does the system of equations below have?\newline−10x+3y=−820x−6y−16=0 \begin{array}{l} -10 x+3 y=-8 \\ 20 x-6 y-16=0 \end{array} \newline(A) no solution\newline(B) one solution\newline(C) infinitely many solutions
  1. Given Equations: We are given the system of equations:\newline−10x+3y=−8-10x + 3y = -8\newline20x−6y=1620x - 6y = 16\newlineFirst, let's simplify the second equation by dividing all terms by 22 to make it easier to compare with the first equation.\newline20x2−6y2=162\frac{20x}{2} - \frac{6y}{2} = \frac{16}{2}\newline10x−3y=810x - 3y = 8
  2. Simplify Second Equation: Now we have the system of equations:\newline−10x+3y=−8-10x + 3y = -8\newline10x−3y=810x - 3y = 8\newlineWe can add the two equations together to see if they are consistent or inconsistent.\newline(−10x+3y)+(10x−3y)=−8+8(-10x + 3y) + (10x - 3y) = -8 + 8\newline−10x+10x+3y−3y=0-10x + 10x + 3y - 3y = 0\newline0=00 = 0
  3. Add Equations Together: Since the left-hand side of the equation simplifies to 00 and the right-hand side is also 00, the equations are dependent, meaning they represent the same line.\newlineTherefore, the system of equations has infinitely many solutions.

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